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Chords of diaschismic

Edit 9 2 …
Below are listed the dyadic chords of 11-limit diaschismic temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 441/440 are labeled werckismic, by 176/175 valinorsmic, by 896/891 pentacircle, and by 126/125 starling. Those requiring tempering by any two of 126/125, 176/175 or 441/440 are labeled thrush, and by both 896/891 and 441/440 pele.

The normal mapping for diaschismic is dia = [<2 0 11 31 45|, <0 1 -2 -8 -12|]. From this we may derive a val v = dia[1] - 100 dia[2] = <2 -100 211 831 1245| which we may use to sort and normalize the chords of diaschismic. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v. If we look at the highest, rightmost, element of the chord, divide that by 100, round, and multiply by 2, we get the Graham complexity of the chord. Redundantly for the sake of convenience, the Graham complexity is listed in the last column.

Diaschsmic has MOS of size 10, 12, 22, 34, 46 and 58. It may be seen that even ten notes are not without resources, mostly essentially just. By the time we get to 22 notes, we are well-supplied.

Triads

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-104-208
 1-4/3-16/9
 ambitonal
 4
2
 0-104-311
 1-4/3-5/3
 otonal
 6
3
 0-207-311
 1-5/4-5/3
 utonal
 6
4
 0-104-413
 1-4/3-10/9
 otonal
 8
5
 0-207-413
 1-5/4-10/9
 utonal
 8
6
 0-208-413
 1-16/9-10/9
 otonal
 8
7
 0-311-413
 1-5/3-10/9
 utonal
 8
8
 0-207-414
 1-5/4-11/7
 valinorsmic
 8
9
 0-208-414
 1-16/9-11/7
 pentacircle
 8
10
 0-207-620
 1-5/4-7/5
 starling
 12
11
 0-208-620
 1-16/9-7/5
 werckismic
 12
12
 0-311-620
 1-5/3-7/5
 starling
 12
13
 0-413-620
 1-10/9-7/5
 starling
 12
14
 0-414-620
 1-11/7-7/5
 werckismic
 12
15
 0-207-827
 1-5/4-7/4
 otonal
 16
16
 0-413-827
 1-10/9-7/4
 werckismic
 16
17
 0-414-827
 1-11/7-7/4
 werckismic
 16
18
 0-620-827
 1-7/5-7/4
 utonal
 16
19
 0-104-929
 1-4/3-7/6
 otonal
 18
20
 0-311-929
 1-5/3-7/6
 otonal
 18
21
 0-620-929
 1-7/5-7/6
 utonal
 18
22
 0-827-929
 1-7/4-7/6
 utonal
 18
23
 0-207-1032
 1-5/4-11/10
 valinorsmic
 20
24
 0-414-1032
 1-11/7-11/10
 utonal
 20
25
 0-620-1032
 1-7/5-11/10
 otonal
 20
26
 0-827-1032
 1-7/4-11/10
 valinorsmic
 20
27
 0-104-1033
 1-4/3-14/9
 otonal
 20
28
 0-208-1033
 1-16/9-14/9
 otonal
 20
29
 0-413-1033
 1-10/9-14/9
 otonal
 20
30
 0-620-1033
 1-7/5-14/9
 utonal
 20
31
 0-827-1033
 1-7/4-14/9
 utonal
 20
32
 0-929-1033
 1-7/6-14/9
 utonal
 20
33
 0-207-1239
 1-5/4-11/8
 otonal
 24
34
 0-208-1239
 1-16/9-11/8
 pentacircle
 24
35
 0-414-1239
 1-11/7-11/8
 utonal
 24
36
 0-827-1239
 1-7/4-11/8
 otonal
 24
37
 0-1032-1239
 1-11/10-11/8
 utonal
 24
38
 0-1033-1239
 1-14/9-11/8
 pentacircle
 24
39
 0-104-1343
 1-4/3-11/6
 otonal
 26
40
 0-311-1343
 1-5/3-11/6
 otonal
 26
41
 0-414-1343
 1-11/7-11/6
 utonal
 26
42
 0-929-1343
 1-7/6-11/6
 otonal
 26
43
 0-1032-1343
 1-11/10-11/6
 utonal
 26
44
 0-1239-1343
 1-11/8-11/6
 utonal
 26
45
 0-104-1445
 1-4/3-11/9
 otonal
 28
46
 0-208-1445
 1-16/9-11/9
 otonal
 28
47
 0-413-1445
 1-10/9-11/9
 otonal
 28
48
 0-414-1445
 1-11/7-11/9
 utonal
 28
49
 0-620-1445
 1-7/5-11/9
 werckismic
 28
50
 0-827-1445
 1-7/4-11/9
 werckismic
 28
51
 0-1032-1445
 1-11/10-11/9
 utonal
 28
52
 0-1033-1445
 1-14/9-11/9
 otonal
 28
53
 0-1239-1445
 1-11/8-11/9
 utonal
 28
54
 0-1343-1445
 1-11/6-11/9
 utonal
 28

Tetrads

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-104-208-413
 1-4/3-16/9-10/9
 otonal
 8
2
 0-104-311-413
 1-4/3-5/3-10/9
 ambitonal
 8
3
 0-207-311-413
 1-5/4-5/3-10/9
 utonal
 8
4
 0-207-311-620
 1-5/4-5/3-7/5
 starling
 12
5
 0-207-413-620
 1-5/4-10/9-7/5
 starling
 12
6
 0-208-413-620
 1-16/9-10/9-7/5
 thrush
 12
7
 0-311-413-620
 1-5/3-10/9-7/5
 starling
 12
8
 0-207-414-620
 1-5/4-11/7-7/5
 thrush
 12
9
 0-208-414-620
 1-16/9-11/7-7/5
 pele
 12
10
 0-207-413-827
 1-5/4-10/9-7/4
 werckismic
 16
11
 0-207-414-827
 1-5/4-11/7-7/4
 thrush
 16
12
 0-207-620-827
 1-5/4-7/5-7/4
 starling
 16
13
 0-413-620-827
 1-10/9-7/5-7/4
 thrush
 16
14
 0-414-620-827
 1-11/7-7/5-7/4
 werckismic
 16
15
 0-104-311-929
 1-4/3-5/3-7/6
 otonal
 18
16
 0-311-620-929
 1-5/3-7/5-7/6
 starling
 18
17
 0-620-827-929
 1-7/5-7/4-7/6
 utonal
 18
18
 0-207-414-1032
 1-5/4-11/7-11/10
 valinorsmic
 20
19
 0-207-620-1032
 1-5/4-7/5-11/10
 thrush
 20
20
 0-414-620-1032
 1-11/7-7/5-11/10
 werckismic
 20
21
 0-207-827-1032
 1-5/4-7/4-11/10
 valinorsmic
 20
22
 0-414-827-1032
 1-11/7-7/4-11/10
 thrush
 20
23
 0-620-827-1032
 1-7/5-7/4-11/10
 valinorsmic
 20
24
 0-104-208-1033
 1-4/3-16/9-14/9
 otonal
 20
25
 0-104-413-1033
 1-4/3-10/9-14/9
 otonal
 20
26
 0-208-413-1033
 1-16/9-10/9-14/9
 otonal
 20
27
 0-208-620-1033
 1-16/9-7/5-14/9
 werckismic
 20
28
 0-413-620-1033
 1-10/9-7/5-14/9
 starling
 20
29
 0-413-827-1033
 1-10/9-7/4-14/9
 werckismic
 20
30
 0-620-827-1033
 1-7/5-7/4-14/9
 utonal
 20
31
 0-104-929-1033
 1-4/3-7/6-14/9
 ambitonal
 20
32
 0-620-929-1033
 1-7/5-7/6-14/9
 utonal
 20
33
 0-827-929-1033
 1-7/4-7/6-14/9
 utonal
 20
34
 0-207-414-1239
 1-5/4-11/7-11/8
 valinorsmic
 24
35
 0-208-414-1239
 1-16/9-11/7-11/8
 pentacircle
 24
36
 0-207-827-1239
 1-5/4-7/4-11/8
 otonal
 24
37
 0-414-827-1239
 1-11/7-7/4-11/8
 werckismic
 24
38
 0-207-1032-1239
 1-5/4-11/10-11/8
 valinorsmic
 24
39
 0-414-1032-1239
 1-11/7-11/10-11/8
 utonal
 24
40
 0-827-1032-1239
 1-7/4-11/10-11/8
 valinorsmic
 24
41
 0-208-1033-1239
 1-16/9-14/9-11/8
 pentacircle
 24
42
 0-827-1033-1239
 1-7/4-14/9-11/8
 pentacircle
 24
43
 0-104-311-1343
 1-4/3-5/3-11/6
 otonal
 26
44
 0-104-929-1343
 1-4/3-7/6-11/6
 otonal
 26
45
 0-311-929-1343
 1-5/3-7/6-11/6
 otonal
 26
46
 0-414-1032-1343
 1-11/7-11/10-11/6
 utonal
 26
47
 0-414-1239-1343
 1-11/7-11/8-11/6
 utonal
 26
48
 0-1032-1239-1343
 1-11/10-11/8-11/6
 utonal
 26
49
 0-104-208-1445
 1-4/3-16/9-11/9
 otonal
 28
50
 0-104-413-1445
 1-4/3-10/9-11/9
 otonal
 28
51
 0-208-413-1445
 1-16/9-10/9-11/9
 otonal
 28
52
 0-208-414-1445
 1-16/9-11/7-11/9
 pentacircle
 28
53
 0-208-620-1445
 1-16/9-7/5-11/9
 werckismic
 28
54
 0-413-620-1445
 1-10/9-7/5-11/9
 thrush
 28
55
 0-414-620-1445
 1-11/7-7/5-11/9
 werckismic
 28
56
 0-413-827-1445
 1-10/9-7/4-11/9
 werckismic
 28
57
 0-414-827-1445
 1-11/7-7/4-11/9
 werckismic
 28
58
 0-620-827-1445
 1-7/5-7/4-11/9
 werckismic
 28
59
 0-414-1032-1445
 1-11/7-11/10-11/9
 utonal
 28
60
 0-620-1032-1445
 1-7/5-11/10-11/9
 werckismic
 28
61
 0-827-1032-1445
 1-7/4-11/10-11/9
 thrush
 28
62
 0-104-1033-1445
 1-4/3-14/9-11/9
 otonal
 28
63
 0-208-1033-1445
 1-16/9-14/9-11/9
 otonal
 28
64
 0-413-1033-1445
 1-10/9-14/9-11/9
 otonal
 28
65
 0-620-1033-1445
 1-7/5-14/9-11/9
 werckismic
 28
66
 0-827-1033-1445
 1-7/4-14/9-11/9
 werckismic
 28
67
 0-208-1239-1445
 1-16/9-11/8-11/9
 pentacircle
 28
68
 0-414-1239-1445
 1-11/7-11/8-11/9
 utonal
 28
69
 0-827-1239-1445
 1-7/4-11/8-11/9
 werckismic
 28
70
 0-1032-1239-1445
 1-11/10-11/8-11/9
 utonal
 28
71
 0-1033-1239-1445
 1-14/9-11/8-11/9
 pentacircle
 28
72
 0-104-1343-1445
 1-4/3-11/6-11/9
 ambitonal
 28
73
 0-414-1343-1445
 1-11/7-11/6-11/9
 utonal
 28
74
 0-1032-1343-1445
 1-11/10-11/6-11/9
 utonal
 28
75
 0-1239-1343-1445
 1-11/8-11/6-11/9
 utonal
 28

Pentads

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-207-311-413-620
 1-5/4-5/3-10/9-7/5
 starling
 12
2
 0-207-413-620-827
 1-5/4-10/9-7/5-7/4
 thrush
 16
3
 0-207-414-620-827
 1-5/4-11/7-7/5-7/4
 thrush
 16
4
 0-207-414-620-1032
 1-5/4-11/7-7/5-11/10
 thrush
 20
5
 0-207-414-827-1032
 1-5/4-11/7-7/4-11/10
 thrush
 20
6
 0-207-620-827-1032
 1-5/4-7/5-7/4-11/10
 thrush
 20
7
 0-414-620-827-1032
 1-11/7-7/5-7/4-11/10
 thrush
 20
8
 0-104-208-413-1033
 1-4/3-16/9-10/9-14/9
 otonal
 20
9
 0-208-413-620-1033
 1-16/9-10/9-7/5-14/9
 thrush
 20
10
 0-413-620-827-1033
 1-10/9-7/5-7/4-14/9
 thrush
 20
11
 0-620-827-929-1033
 1-7/5-7/4-7/6-14/9
 utonal
 20
12
 0-207-414-827-1239
 1-5/4-11/7-7/4-11/8
 thrush
 24
13
 0-207-414-1032-1239
 1-5/4-11/7-11/10-11/8
 valinorsmic
 24
14
 0-207-827-1032-1239
 1-5/4-7/4-11/10-11/8
 valinorsmic
 24
15
 0-414-827-1032-1239
 1-11/7-7/4-11/10-11/8
 thrush
 24
16
 0-104-311-929-1343
 1-4/3-5/3-7/6-11/6
 otonal
 26
17
 0-414-1032-1239-1343
 1-11/7-11/10-11/8-11/6
 utonal
 26
18
 0-104-208-413-1445
 1-4/3-16/9-10/9-11/9
 otonal
 28
19
 0-208-413-620-1445
 1-16/9-10/9-7/5-11/9
 thrush
 28
20
 0-208-414-620-1445
 1-16/9-11/7-7/5-11/9
 pele
 28
21
 0-413-620-827-1445
 1-10/9-7/5-7/4-11/9
 thrush
 28
22
 0-414-620-827-1445
 1-11/7-7/5-7/4-11/9
 werckismic
 28
23
 0-414-620-1032-1445
 1-11/7-7/5-11/10-11/9
 werckismic
 28
24
 0-414-827-1032-1445
 1-11/7-7/4-11/10-11/9
 thrush
 28
25
 0-620-827-1032-1445
 1-7/5-7/4-11/10-11/9
 thrush
 28
26
 0-104-208-1033-1445
 1-4/3-16/9-14/9-11/9
 otonal
 28
27
 0-104-413-1033-1445
 1-4/3-10/9-14/9-11/9
 otonal
 28
28
 0-208-413-1033-1445
 1-16/9-10/9-14/9-11/9
 otonal
 28
29
 0-208-620-1033-1445
 1-16/9-7/5-14/9-11/9
 werckismic
 28
30
 0-413-620-1033-1445
 1-10/9-7/5-14/9-11/9
 thrush
 28
31
 0-413-827-1033-1445
 1-10/9-7/4-14/9-11/9
 werckismic
 28
32
 0-620-827-1033-1445
 1-7/5-7/4-14/9-11/9
 werckismic
 28
33
 0-208-414-1239-1445
 1-16/9-11/7-11/8-11/9
 pentacircle
 28
34
 0-414-827-1239-1445
 1-11/7-7/4-11/8-11/9
 werckismic
 28
35
 0-414-1032-1239-1445
 1-11/7-11/10-11/8-11/9
 utonal
 28
36
 0-827-1032-1239-1445
 1-7/4-11/10-11/8-11/9
 thrush
 28
37
 0-208-1033-1239-1445
 1-16/9-14/9-11/8-11/9
 pentacircle
 28
38
 0-827-1033-1239-1445
 1-7/4-14/9-11/8-11/9
 pele
 28
39
 0-414-1032-1343-1445
 1-11/7-11/10-11/6-11/9
 utonal
 28
40
 0-414-1239-1343-1445
 1-11/7-11/8-11/6-11/9
 utonal
 28
41
 0-1032-1239-1343-1445
 1-11/10-11/8-11/6-11/9
 utonal
 28

Hexads

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-207-414-620-827-1032
 1-5/4-11/7-7/5-7/4-11/10
 thrush
 20
2
 0-207-414-827-1032-1239
 1-5/4-11/7-7/4-11/10-11/8
 thrush
 24
3
 0-414-620-827-1032-1445
 1-11/7-7/5-7/4-11/10-11/9
 thrush
 28
4
 0-104-208-413-1033-1445
 1-4/3-16/9-10/9-14/9-11/9
 otonal
 28
5
 0-208-413-620-1033-1445
 1-16/9-10/9-7/5-14/9-11/9
 thrush
 28
6
 0-413-620-827-1033-1445
 1-10/9-7/5-7/4-14/9-11/9
 thrush
 28
7
 0-414-827-1032-1239-1445
 1-11/7-7/4-11/10-11/8-11/9
 thrush
 28
8
 0-414-1032-1239-1343-1445
 1-11/7-11/10-11/8-11/6-11/9
 utonal
 28